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- 1. Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Introduction to Quantum Computation Part - II Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Ritajit Majumdar, Arunabha Saha Reference University of Calcutta September 25, 2013 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 1 / 54
- 2. Introduction to Quantum Computation 1 2 Ritajit Majumdar, Arunabha Saha Introduction EPR Paradox Outline Introduction EPR Paradox Bell’s Theorem 3 Bell’s Theorem Mathematical Notation Quantum Gates 4 Mathematical Notation Super Dense Coding Next Presentation 5 6 Super Dense Coding 7 Next Presentation 8 Reference Quantum Gates Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 2 / 54
- 3. Introduction to Quantum Computation The Confusion Ritajit Majumdar, Arunabha Saha Outline Introduction What exactly a quantum state means??!! EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 3 / 54
- 4. Introduction to Quantum Computation The Confusion Ritajit Majumdar, Arunabha Saha Outline Introduction What exactly a quantum state means??!! EPR Paradox Bell’s Theorem |ψ does not uniquely determine the outcome of measurement. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 3 / 54
- 5. Introduction to Quantum Computation The Confusion Ritajit Majumdar, Arunabha Saha Outline Introduction What exactly a quantum state means??!! EPR Paradox Bell’s Theorem |ψ does not uniquely determine the outcome of measurement. Mathematical Notation Quantum Gates Super Dense Coding It provides the statistical distribution of all possible outcomes. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Next Presentation Reference September 25, 2013 3 / 54
- 6. Introduction to Quantum Computation The Confusion Ritajit Majumdar, Arunabha Saha Outline Introduction What exactly a quantum state means??!! EPR Paradox Bell’s Theorem |ψ does not uniquely determine the outcome of measurement. Mathematical Notation Quantum Gates Super Dense Coding It provides the statistical distribution of all possible outcomes. Next Presentation Reference Here arises the confusion Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 3 / 54
- 7. Introduction to Quantum Computation The Confusion Ritajit Majumdar, Arunabha Saha Outline Introduction What exactly a quantum state means??!! EPR Paradox Bell’s Theorem |ψ does not uniquely determine the outcome of measurement. Mathematical Notation Quantum Gates Super Dense Coding It provides the statistical distribution of all possible outcomes. Next Presentation Reference Here arises the confusion ? Does the physical system ‘actually have’ the attributes prior to measurement (realist viewpoint) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 3 / 54
- 8. Introduction to Quantum Computation The Confusion Ritajit Majumdar, Arunabha Saha Outline Introduction What exactly a quantum state means??!! EPR Paradox Bell’s Theorem |ψ does not uniquely determine the outcome of measurement. Mathematical Notation Quantum Gates Super Dense Coding It provides the statistical distribution of all possible outcomes. Next Presentation Reference Here arises the confusion ? Does the physical system ‘actually have’ the attributes prior to measurement (realist viewpoint) OR ? The properties are ‘created’ by measurement (orthodox position) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 3 / 54
- 9. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference EPR Paradox Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 4 / 54
- 10. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha 0 − Pi-meson decay: π → e + e + Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Singlet state: |ψ = |01 −|10 √ 2 Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 5 / 54
- 11. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha 0 − Pi-meson decay: π → e + e + Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Singlet state: |ψ = |01 −|10 √ 2 Quantum Gates Super Dense Coding Next Presentation If the electron (e − ) is found spin up (or |0 ) then the positron (e + ) will be found to be spin down (or |1 ) and vice-versa. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 25, 2013 5 / 54
- 12. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha 0 − Pi-meson decay: π → e + e + Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Singlet state: |ψ = |01 −|10 √ 2 Quantum Gates Super Dense Coding Next Presentation If the electron (e − ) is found spin up (or |0 ) then the positron (e + ) will be found to be spin down (or |1 ) and vice-versa. Reference Quantum mechanics does not ensure which combination will be obtained but it is observed that the measurement is correlated. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 5 / 54
- 13. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha 0 − Pi-meson decay: π → e + e + Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Singlet state: |ψ = |01 −|10 √ 2 Quantum Gates Super Dense Coding Next Presentation If the electron (e − ) is found spin up (or |0 ) then the positron (e + ) will be found to be spin down (or |1 ) and vice-versa. Reference Quantum mechanics does not ensure which combination will be obtained but it is observed that the measurement is correlated. 1 Each combination is obtained with probability 2 . But if the electron is found to be in state |0 then the positron is deﬁnitely in state |1 and vice versa. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 5 / 54
- 14. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha Outline This correlation of measurement is independent of spatial distance. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 6 / 54
- 15. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha Outline This correlation of measurement is independent of spatial distance. Introduction EPR Paradox Bell’s Theorem One school of thought claimed that the particle has neither spin up nor spin down prior to measurement, it is just created by the act of measurement. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 6 / 54
- 16. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha Outline This correlation of measurement is independent of spatial distance. Introduction EPR Paradox Bell’s Theorem One school of thought claimed that the particle has neither spin up nor spin down prior to measurement, it is just created by the act of measurement. Einstein mentioned this as “spooky action at a distance”. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference September 25, 2013 6 / 54
- 17. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha Outline This correlation of measurement is independent of spatial distance. Introduction EPR Paradox Bell’s Theorem One school of thought claimed that the particle has neither spin up nor spin down prior to measurement, it is just created by the act of measurement. Einstein mentioned this as “spooky action at a distance”. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference EPR argument is based on principle of locality. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 6 / 54
- 18. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha Outline This correlation of measurement is independent of spatial distance. Introduction EPR Paradox Bell’s Theorem One school of thought claimed that the particle has neither spin up nor spin down prior to measurement, it is just created by the act of measurement. Einstein mentioned this as “spooky action at a distance”. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference EPR argument is based on principle of locality. Principle of Locality No inﬂuence can propagate faster than light. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 6 / 54
- 19. Introduction to Quantum Computation EPR Paradox Ritajit Majumdar, Arunabha Saha Outline This correlation of measurement is independent of spatial distance. Introduction EPR Paradox Bell’s Theorem One school of thought claimed that the particle has neither spin up nor spin down prior to measurement, it is just created by the act of measurement. Einstein mentioned this as “spooky action at a distance”. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference EPR argument is based on principle of locality. Principle of Locality No inﬂuence can propagate faster than light. But if we claim that the collapse is not instantaneous, then it leads to violation of angular momentum conservation.(Pi-meson decay) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 6 / 54
- 20. Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Bell‘s Theorem Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 7 / 54
- 21. Local Hidden Variable Theory (LHVT) Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR paper (Phys. Rev. 47, 777 (1935)) In quantum mechanics, in the case of two physical quantities described by non-commutating operators, the knowledge of one precludes the knowledge of another. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete, or (2) these two quantities cannot have simultaneous reality. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference September 25, 2013 8 / 54
- 22. Local Hidden Variable Theory (LHVT) Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR paper (Phys. Rev. 47, 777 (1935)) In quantum mechanics, in the case of two physical quantities described by non-commutating operators, the knowledge of one precludes the knowledge of another. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete, or (2) these two quantities cannot have simultaneous reality. EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference The wavefunction, ψ, does not describe the system fully. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 8 / 54
- 23. Local Hidden Variable Theory (LHVT) Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR paper (Phys. Rev. 47, 777 (1935)) In quantum mechanics, in the case of two physical quantities described by non-commutating operators, the knowledge of one precludes the knowledge of another. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete, or (2) these two quantities cannot have simultaneous reality. EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference The wavefunction, ψ, does not describe the system fully. Another quantity (say λ) addition to ψ is needed to describe the system completely. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 8 / 54
- 24. Local Hidden Variable Theory (LHVT) Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR paper (Phys. Rev. 47, 777 (1935)) In quantum mechanics, in the case of two physical quantities described by non-commutating operators, the knowledge of one precludes the knowledge of another. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete, or (2) these two quantities cannot have simultaneous reality. EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference The wavefunction, ψ, does not describe the system fully. Another quantity (say λ) addition to ψ is needed to describe the system completely. This λ is called “hidden variable” - we have no idea how to calculate or measure it. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 8 / 54
- 25. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha In 1964 Bell proved that any LHVT is incompatible with quantum mechanics. [J.S. Bell, physics 1, 195(1964)] Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 9 / 54
- 26. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha In 1964 Bell proved that any LHVT is incompatible with quantum mechanics. [J.S. Bell, physics 1, 195(1964)] Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Charlie prepares two particles and sends one to Alice and the other to Bob. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 9 / 54
- 27. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha In 1964 Bell proved that any LHVT is incompatible with quantum mechanics. [J.S. Bell, physics 1, 195(1964)] Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Charlie prepares two particles and sends one to Alice and the other to Bob. On getting the particle, Alice can measure physical properties PQ and PR . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 9 / 54
- 28. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha In 1964 Bell proved that any LHVT is incompatible with quantum mechanics. [J.S. Bell, physics 1, 195(1964)] Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Charlie prepares two particles and sends one to Alice and the other to Bob. On getting the particle, Alice can measure physical properties PQ and PR . Alice ﬂips a coin and decides which measurement has to be done i.e. measuring randomly. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 9 / 54
- 29. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Q and R is the value for the property PQ and PR respectively. Each have one of the two outcomes +1 or -1. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 10 / 54
- 30. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Q and R is the value for the property PQ and PR respectively. Each have one of the two outcomes +1 or -1. Similar is for Bob (Hence PS , PT ). Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 10 / 54
- 31. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Q and R is the value for the property PQ and PR respectively. Each have one of the two outcomes +1 or -1. Similar is for Bob (Hence PS , PT ). The experiment is arranged so that Alice and Bob can perform measurements at the same time (more precisely in a causally disconnected manner). Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference September 25, 2013 10 / 54
- 32. Introduction to Quantum Computation Bell‘s Theorem Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Q and R is the value for the property PQ and PR respectively. Each have one of the two outcomes +1 or -1. Similar is for Bob (Hence PS , PT ). The experiment is arranged so that Alice and Bob can perform measurements at the same time (more precisely in a causally disconnected manner). Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference By locality principle Alice’s measurement cannot disturb Bob‘s measurement. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 10 / 54
- 33. Introduction to Quantum Computation CHSH Inequality Ritajit Majumdar, Arunabha Saha Now perform simple algebra with the quantity: Outline Introduction QS + RS + RT - QT EPR Paradox Bell’s Theorem QS + RS + RT − QT = (Q + R)S + (R − Q)T Since R, Q = ±1, it follows that either QS + RS + RT − QT = 0 In either case (Q + R)S + (R − Q)T = ±2 Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Let p(q, r , s, t) is the probability that, before the measurements are performed, the system is in state Q = q, R = r , S = s, T = t Let E(.) denote the mean value. E(QS + RS + RT − QT ) = Σqrst p(q, r , s, t)(qs + rs + rt − qt) Σqrst p(q, r , s, t)x2 =2 .......(1) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 11 / 54
- 34. Introduction to Quantum Computation CHSH Inequality Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox E(QS + RS + RT − QT ) = Σqrst p(q, r , s, t)qs + Σqrst p(q, r , s, t)rs + Σqrst p(q, r , s, t)rt − Σqrst p(q, r , s, t)qt Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding = E(QS) + E(RS) + E(RT ) − E(QT ) .......(2) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Next Presentation Reference September 25, 2013 12 / 54
- 35. Introduction to Quantum Computation CHSH Inequality Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox E(QS + RS + RT − QT ) = Σqrst p(q, r , s, t)qs + Σqrst p(q, r , s, t)rs + Σqrst p(q, r , s, t)rt − Σqrst p(q, r , s, t)qt Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding = E(QS) + E(RS) + E(RT ) − E(QT ) .......(2) Comparing eqn.(1)and eqn.(2) we obtain Bell Inequality E(QS) + E(RS) + E(RT ) − E(QT ) Next Presentation Reference 2 This result also term as CHSH inequality. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 12 / 54
- 36. Nature Does Not follow Bell Inequality Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Let Charlie prepares a quantum system of two qubits and shared to Alice and Bob |ψ = Introduction EPR Paradox Bell’s Theorem Mathematical Notation |01 −|10 √ 2 Quantum Gates Super Dense Coding After measurement the following observable 2 Q = Z1 S = −Z√−X2 2 R = X1 QS = Next Presentation Reference Z2√ 2 −X 2 1 √ ; 2 T = 1 RS = √2 ; RT = 1 √ 2 ; QT = 1 √ 2 Therefore, √ QS + RS + RT − QT = 2 2 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 13 / 54
- 37. Nature Does Not follow Bell Inequality Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Hence it is observed that the previous result violates Bell inequality. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 14 / 54
- 38. Nature Does Not follow Bell Inequality Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Hence it is observed that the previous result violates Bell inequality. Bell inequality is not obeyed by Nature. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 14 / 54
- 39. Nature Does Not follow Bell Inequality Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Hence it is observed that the previous result violates Bell inequality. Bell inequality is not obeyed by Nature. This indicates that may be some of the basic assumptions are incorrect. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 14 / 54
- 40. Nature Does Not follow Bell Inequality Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Hence it is observed that the previous result violates Bell inequality. Bell inequality is not obeyed by Nature. This indicates that may be some of the basic assumptions are incorrect. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Assumptions of local realism Reference (1) Physical properties PQ , PR , PS , PT have deﬁnite values Q, R, S, T which exist independent of observation. (Assumption of realism) (2) Alice measurement does not inﬂuence the result of Bob‘s measurement. (Assumption of locality ) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 14 / 54
- 41. Introduction to Quantum Computation Bipartite System Ritajit Majumdar, Arunabha Saha Outline Introduction Suppose we have two parties, Alice and Bob, each having a qubit. Alice’s qubit is denoted as |ψ1 = α0 |0 + α1 |1 and that of Bob is denoted as |ψ2 = β0 |0 + β1 |1 . EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates So what will be the state of the composite system of Alice and Bob? Super Dense Coding Next Presentation Reference 1 |0 ⊗ |0 ≡ |0 |0 ≡ |00 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 15 / 54
- 42. Introduction to Quantum Computation Bipartite System Ritajit Majumdar, Arunabha Saha Outline Introduction Suppose we have two parties, Alice and Bob, each having a qubit. Alice’s qubit is denoted as |ψ1 = α0 |0 + α1 |1 and that of Bob is denoted as |ψ2 = β0 |0 + β1 |1 . EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates So what will be the state of the composite system of Alice and Bob? Super Dense Coding Next Presentation Reference It is given as |ψ12 = (α0 |0 + α1 |1 ) ⊗ (β0 |0 + β1 |1 ) = α0 β0 |00 1 |0 1 +α0 β1 |01 + α1 β0 |10 + α1 β1 |11 ⊗ |0 ≡ |0 |0 ≡ |00 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 15 / 54
- 43. Bipartite System (Contd.) Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Similar is the reverse case. If we have a bipartite system denoted as α0 β0 |00 + α0 β1 |01 + α1 β0 |10 + α1 β1 |11 then we can factorise it to get the corresponding qubits involved. Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 16 / 54
- 44. Introduction to Quantum Computation Entangled State Ritajit Majumdar, Arunabha Saha A pure state of two systems is entangled if it cannot be written as a product of two states - Outline Introduction EPR Paradox Bell’s Theorem |ψAB = |ψA ⊗ |ψB Mathematical Notation We have four such entangled states called Bell States or EPR States. Quantum Gates Super Dense Coding Next Presentation Reference |φ+ |φ− |ψ+ |ψ− = = = = 1 √ (|00 2 1 √ (|00 2 1 √ (|01 2 1 √ (|01 2 + |11 − |11 + |10 − |10 ) ) ) ) These 4 states are called Maximally Entangled States. Any state of the form a |00 ± b |11 or a |01 ± b |10 , where a = b, is called Pure Entangled State. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 17 / 54
- 45. Introduction to Quantum Computation Why Can’t be Factorised? Ritajit Majumdar, Arunabha Saha Outline Introduction Let us consider the state |ψ = 1 √ (|00 2 + |11 ) And suppose we can factorise the state in the form EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates (α0 |0 + α1 |1 ) ⊗ (β0 |0 + β1 |1 ). Super Dense Coding Next Presentation Reference So we should have 1 α0 β0 = √2 α0 β1 = 0 α1 β0 = 0 1 α1 β1 = √2 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 18 / 54
- 46. Why Can’t be Factorised? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline From the 2 nd rd and the 3 equations we have - Introduction EPR Paradox Bell’s Theorem either Mathematical Notation α0 = 0 or β1 = 0 AND Quantum Gates Super Dense Coding Next Presentation Reference either α1 = 0 or β0 = 0. But if any two of those have 0 values, then the 1st and the last equations are not satisﬁed. Hence the state cannot be factorised. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 19 / 54
- 47. Measurement of Entangled bits. For entangled states, |ψAB = |ψA ⊗ |ψB Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Consider the state |ψ = 1 √ (|01 2 + |10 ) Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 20 / 54
- 48. Measurement of Entangled bits. For entangled states, |ψAB = |ψA ⊗ |ψB Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Consider the state |ψ = 1 √ (|01 2 + |10 ) Introduction EPR Paradox What happens if we measure the 1st qubit only? Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 20 / 54
- 49. Measurement of Entangled bits. For entangled states, |ψAB = |ψA ⊗ |ψB Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Consider the state |ψ = 1 √ (|01 2 + |10 ) Introduction EPR Paradox What happens if we measure the 1st qubit only? Suppose after measurement, we found the 1st qubit to be in state |0 . Then without measurement, we can know immediately that the 2nd qubit is in state |1 . Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 20 / 54
- 50. Measurement of Entangled bits. For entangled states, |ψAB = |ψA ⊗ |ψB Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Consider the state |ψ = 1 √ (|01 2 + |10 ) Introduction EPR Paradox What happens if we measure the 1st qubit only? Suppose after measurement, we found the 1st qubit to be in state |0 . Then without measurement, we can know immediately that the 2nd qubit is in state |1 . Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference So irrespective of the spatial distance between the two entangled qubits, measuring one of them disturbs the state of the other. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 20 / 54
- 51. Introduction to Quantum Computation Quantum Gates Ritajit Majumdar, Arunabha Saha “When we get to the very, very small world - say circuits of seven atoms - we have a lot of new things that would happen that represent completely new opportunities for design. Atoms on small scale behave like nothing on a large scale, for they satisfy the laws of quantum mechanics. So, as we go down and ﬁddle around with the atoms there, we are working with diﬀerent laws, and we can expect to do diﬀerent things” - Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Richard Feynman Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 25, 2013 21 / 54
- 52. Evolution of Quantum Systems Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Quantum systems evolve by Unitary transformations. If |ψ(t1 ) is the state of the system at time t1 and |ψ(t2 ) is the state of the system at time t2 , then - Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding |ψ(t2 ) = U(t1 , t2 ) |ψ(t1 ) Next Presentation Reference where U(t1 , t2 ) is a Unitary Matrix. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 22 / 54
- 53. Evolution of Quantum Systems Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Quantum systems evolve by Unitary transformations. If |ψ(t1 ) is the state of the system at time t1 and |ψ(t2 ) is the state of the system at time t2 , then - Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding |ψ(t2 ) = U(t1 , t2 ) |ψ(t1 ) Next Presentation Reference where U(t1 , t2 ) is a Unitary Matrix. Analogous to Classical AND, OR, NOT etc. gates, Quantum Computation has its own “Gates” which are represented as Unitary Matrices. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 22 / 54
- 54. Introduction to Quantum Computation Classical Gates Ritajit Majumdar, Arunabha Saha Outline Classical Computers consist of wires that carry bits of informtion and Gates that transform these bits in some way. Introduction EPR Paradox Bell’s Theorem Some of the famous classical logic gates are - Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 23 / 54
- 55. Irreversibility of Classical Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Classical Gates are irreversible 2 , i.e., one cannot determine unique inputs for all outputs. Introduction EPR Paradox Bell’s Theorem For example, in an AND gate if the output is 0, it cannot be determined whether the input values were 00, 01 or 10. Similar is the case for output 1 in OR gate. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference 2 The simplest example of a Classical Reversible Logic Gate is a NOT gate. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 24 / 54
- 56. Irreversibility of Classical Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Classical Gates are irreversible 2 , i.e., one cannot determine unique inputs for all outputs. Introduction EPR Paradox Bell’s Theorem For example, in an AND gate if the output is 0, it cannot be determined whether the input values were 00, 01 or 10. Similar is the case for output 1 in OR gate. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Unfortunately, logical irreversibility comes at a price. Fundamental Physics states that energy must be dissipated when information is erased. And this dissipation is kTln2 per bit erased where k is the Boltzmann constant (k = 1.3805 × 10−23 JK −1 ) and T is the temperature in Absolute Scale. 2 The simplest example of a Classical Reversible Logic Gate is a NOT gate. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 24 / 54
- 57. Introduction to Quantum Computation Quantum Logic Gates Ritajit Majumdar, Arunabha Saha Outline Introduction Just as any classical computation can be broken into a sequence of classical logic gates that act on only a few classical bits at a time, quantum computation too can be broken down into a sequence of quantum logic gates that act on only a few quantum bits at a time. EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference 3 Operation of Quantum Gates on a superposition takes same time as the operation on a basis state. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 25 / 54
- 58. Introduction to Quantum Computation Quantum Logic Gates Ritajit Majumdar, Arunabha Saha Outline Introduction Just as any classical computation can be broken into a sequence of classical logic gates that act on only a few classical bits at a time, quantum computation too can be broken down into a sequence of quantum logic gates that act on only a few quantum bits at a time. EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference The main diﬀerence is that where classical logic gates act on classical bits 0 or 1, quantum gates can manipulate arbitrary multi-partite quantum states including arbitrary superpositions3 of the computational basis states, which may also be entangled. 3 Operation of Quantum Gates on a superposition takes same time as the operation on a basis state. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 25 / 54
- 59. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 26 / 54
- 60. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - Introduction EPR Paradox Bell’s Theorem Mathematical Notation 1. U † is unitary. Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 26 / 54
- 61. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - 2. U EPR Paradox Bell’s Theorem Mathematical Notation 1. U † is unitary. −1 Introduction Quantum Gates is unitary. Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 26 / 54
- 62. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - 2. U Bell’s Theorem Quantum Gates is unitary. Super Dense Coding 3. U −1 = U † (which is the criterion for determining unitarity.) Ritajit Majumdar, Arunabha Saha (CU) EPR Paradox Mathematical Notation 1. U † is unitary. −1 Introduction Introduction to Quantum Computation Next Presentation Reference September 25, 2013 26 / 54
- 63. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - 2. U EPR Paradox Bell’s Theorem Mathematical Notation 1. U † is unitary. −1 Introduction Quantum Gates is unitary. Super Dense Coding 3. U −1 = U † (which is the criterion for determining unitarity.) Next Presentation Reference † 4. UU = I Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 26 / 54
- 64. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - 2. U EPR Paradox Bell’s Theorem Mathematical Notation 1. U † is unitary. −1 Introduction Quantum Gates is unitary. Super Dense Coding 3. U −1 = U † (which is the criterion for determining unitarity.) Next Presentation Reference † 4. UU = I 5. |det(U)| = 1 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 26 / 54
- 65. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - 2. U EPR Paradox Bell’s Theorem Mathematical Notation 1. U † is unitary. −1 Introduction Quantum Gates is unitary. Super Dense Coding 3. U −1 = U † (which is the criterion for determining unitarity.) Next Presentation Reference † 4. UU = I 5. |det(U)| = 1 6. The columns (rows) of U form an orthonormal set of vectors. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 26 / 54
- 66. Properties of Quantum Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The properties of quantum logic gates are the direct consequence that they are described by Unitary Matrices. If U is a Unitary , then the following facts holds - 2. U EPR Paradox Bell’s Theorem Mathematical Notation 1. U † is unitary. −1 Introduction Quantum Gates is unitary. Super Dense Coding 3. U −1 = U † (which is the criterion for determining unitarity.) Next Presentation Reference † 4. UU = I 5. |det(U)| = 1 6. The columns (rows) of U form an orthonormal set of vectors. The fact that for any quantum gate U, UU † = I , ensures that we can always undo a quantum gate, i.e, a quantum gate is logically reversible. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 26 / 54
- 67. Notation of Quantum Logic Gates Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Just like Classical computation, in quantum we consider a quantum input wire that carries a qubit, a quantum gate that performs some transformation on it, and a qauntum output wire 4 that carries the qubit out. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference 4 It may be noted that unlike classical computation, creating a quantum wire is extremely diﬃcult since a qubit evolves with time according to the Schrodinger Equation. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 27 / 54
- 68. Introduction to Quantum Computation Bit Flip Ritajit Majumdar, Arunabha Saha The simplest quantum gate is the “bit ﬂip” gate, which is analogous to the classical NOT gate. Bit Flip is given by the pauli X matrix - Outline Introduction EPR Paradox Bell’s Theorem 0 1 1 0 Mathematical Notation Quantum Gates Super Dense Coding The operation of the Bit Flip gate is - Next Presentation Reference X |0 = |1 X |1 = |0 Figure : Working of a Quantum Bit Flip gate It may be checked that X † = X and XX † = I Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 28 / 54
- 69. Introduction to Quantum Computation Phase Shift Ritajit Majumdar, Arunabha Saha The pauli matrix Z is called the “Phase Shift” gate. The operation of this gate is - Outline Introduction EPR Paradox Z |0 = |0 Z |1 = − |1 Bell’s Theorem Mathematical Notation Quantum Gates The matrix notation of the Phase Flip gate is - Super Dense Coding Next Presentation 1 0 0 −1 Reference Figure : Working of a Quantum Phase Flip gate It may be checked that Z † = Z and ZZ † = I Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 29 / 54
- 70. Introduction to Quantum Computation Hadamard Gate Ritajit Majumdar, Arunabha Saha One of the most important single qubit gate is the Hadamard Gate. The matrix notation of this gate is 1 √ 2 1 √ 2 1 √ 2 1 − √2 Outline Introduction EPR Paradox = 1 √ 2 1 1 1 −1 Bell’s Theorem Mathematical Notation Quantum Gates Hadamard Gate switches between the bit basis and the sign basis. Super Dense Coding Next Presentation Reference H |0 = H |1 = 1 √ 2 1 √ 2 |0 + |0 − 1 √ 2 1 √ 2 |1 = |+ |1 = |− H |+ = |0 H |− = |1 Note that, starting with only the state |0 , Hadamard Trasform can produce an equal superposition of both |0 and |1 . This is an extremely powerful property and is the source of quantum parallelism. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 30 / 54
- 71. Introduction to Quantum Computation Taking Stock Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Figure : A relative view of Bit Flip, Phase Flip and Hadamard Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 31 / 54
- 72. Geometrical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha A unitary transformation is mathematically a rotation. Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 32 / 54
- 73. Geometrical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha A unitary transformation is mathematically a rotation. For the Bit Flip gate X, we have X |0 = |1 and X |1 = |0 . So the bit ﬂip gate can be geometrically thought of as a rotation about an axis which is at an angle of π/4 from the two orthonormal axes |0 and |1 . Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 32 / 54
- 74. Geometrical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha A unitary transformation is mathematically a rotation. For the Bit Flip gate X, we have X |0 = |1 and X |1 = |0 . So the bit ﬂip gate can be geometrically thought of as a rotation about an axis which is at an angle of π/4 from the two orthonormal axes |0 and |1 . Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Figure : Geometrical interpretation of Bit Flip gate Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 32 / 54
- 75. Geometrical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha The geometrical picture of the 3 single qubit gates discussed earlier - Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Figure : The Geometric Interpretation of Bit Flip, Phase Flip and Hadamard Gates Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 33 / 54
- 76. Introduction to Quantum Computation Multiple Qubit Gates Ritajit Majumdar, Arunabha Saha Outline “In natural science, Nature has given us a world and we’re just to discover its laws. In computers, we can stuﬀ laws into it and create a world.” - Alan Kay Introduction EPR Paradox Bell’s Theorem Mathematical Notation The last few gates were single qubit quantum gates. Now we look into few multiple qubit quantum gates. Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 34 / 54
- 77. Introduction to Quantum Computation CNOT Gate A reversible gate of considerable importance in quantum computation is the 2-bit Controlled-NOT (CNOT) gate. The eﬀect of CNOT gate is to ﬂip the 2nd bit if and only if the 1st bit is set to 1. Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation That is, the decision to negate is controlled by the value of the 1st bit. Hence the name. Quantum Gates Super Dense Coding Next Presentation The symbolic representation of CNOT is - Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 25, 2013 35 / 54
- 78. Introduction to Quantum Computation CNOT Gate Ritajit Majumdar, Arunabha Saha The matrix representation of CNOT gate is 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation The truth table of CNOT gate is given as a 0 0 1 1 b 0 1 0 1 a’ 0 0 1 1 Reference b’ 0 1 1 0 CNOT gate is extremely important because it can create Entanglement. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 36 / 54
- 79. Introduction to Quantum Computation Fredkin Gate Ritajit Majumdar, Arunabha Saha Outline Fredkin gate, also called the Controlled-Swap (C-SWAP) gate, is a multiple qubit gate. The action of this gate swaps the 2nd and 3rd qubits only if the 1st qubit i.e. the control bit is 1. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 37 / 54
- 80. Introduction to Quantum Computation Fredkin Gate Ritajit Majumdar, Arunabha Saha Outline Fredkin gate, also called the Controlled-Swap (C-SWAP) gate, is a multiple qubit gate. The action of this gate swaps the 2nd and 3rd qubits only if the 1st qubit i.e. the control bit is 1. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates The matrix representation of the Fredkin Gate is 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Super Dense Coding Next Presentation Reference September 25, 2013 37 / 54
- 81. Introduction to Quantum Computation Fredkin Gate Ritajit Majumdar, Arunabha Saha Outline Fredkin Gate may be treated like a reversible AND gate. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 38 / 54
- 82. Introduction to Quantum Computation Fredkin Gate Ritajit Majumdar, Arunabha Saha Outline Fredkin Gate may be treated like a reversible AND gate. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference From the ﬁgure, it is clear that if C = 0, then the last output gives AB. The 2nd output is then a junk. The 1st input, A, is retained and by operating the gate again, the 2nd input, B, is retrieved. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 38 / 54
- 83. Introduction to Quantum Computation Fredkin Gate Ritajit Majumdar, Arunabha Saha Outline Fredkin Gate may be treated like a reversible AND gate. Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference From the ﬁgure, it is clear that if C = 0, then the last output gives AB. The 2nd output is then a junk. The 1st input, A, is retained and by operating the gate again, the 2nd input, B, is retrieved. Hence it operates like a reversible AND gate. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 38 / 54
- 84. Introduction to Quantum Computation Toﬀoli Gate Ritajit Majumdar, Arunabha Saha Toﬀoli Gate, or the Controlled Controlled NOT (CCNOT) Gate, has 3 input bits and 3 output bits. Two of the bits are control bits that are unaﬀected by the action of the Toﬀoli Gate. The 3rd bit is the target bit which is ﬂipped if both the control bits are set to 1, otherwise left unchanged. Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 39 / 54
- 85. Introduction to Quantum Computation Toﬀoli Gate Ritajit Majumdar, Arunabha Saha Toﬀoli Gate, or the Controlled Controlled NOT (CCNOT) Gate, has 3 input bits and 3 output bits. Two of the bits are control bits that are unaﬀected by the action of the Toﬀoli Gate. The 3rd bit is the target bit which is ﬂipped if both the control bits are set to 1, otherwise left unchanged. Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding The matrix representation of Toﬀoli gate is 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Next Presentation Reference September 25, 2013 39 / 54
- 86. Introduction to Quantum Computation Toﬀoli Gate Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Figure : Toﬀoli Gate: a = a, b = b, c = c ⊕ ab Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 40 / 54
- 87. Introduction to Quantum Computation Toﬀoli Gate Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Figure : Toﬀoli Gate: a = a, b = b, c = c ⊕ ab Next Presentation Reference From the ﬁgure, it is clear that if c = 1, then c = 1 ⊕ ab = ¬(ab) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 40 / 54
- 88. Introduction to Quantum Computation Toﬀoli Gate Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Figure : Toﬀoli Gate: a = a, b = b, c = c ⊕ ab Next Presentation Reference From the ﬁgure, it is clear that if c = 1, then c = 1 ⊕ ab = ¬(ab) Thus Toﬀoli Gate acts as a reversible NAND gate. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 40 / 54
- 89. Introduction to Quantum Computation Toﬀoli Gate Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Figure : Toﬀoli Gate: a = a, b = b, c = c ⊕ ab Next Presentation Reference From the ﬁgure, it is clear that if c = 1, then c = 1 ⊕ ab = ¬(ab) Thus Toﬀoli Gate acts as a reversible NAND gate. Since NAND is a universal gate in Classical Computation, and it can be realised in Quantum Computers by the Toﬀoli Gate, it can be claimed that Any computation that is possible in Classical Computers is also possible in Quantum Computers. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 40 / 54
- 90. Introduction to Quantum Computation Super Dense Coding Ritajit Majumdar, Arunabha Saha Outline Introduction Breaking news! A single qubit can transmit two full classical bits of information. EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 41 / 54
- 91. Quantum Measurement Revisited Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox From the measurement principle of Quantum Mechanics, we know that if we have a state |ψ = α |0 + β |1 then after measurement the state of the system collapses to |0 with probability |α|2 or to |1 with probability |β|2 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference September 25, 2013 42 / 54
- 92. Quantum Measurement Revisited Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox From the measurement principle of Quantum Mechanics, we know that if we have a state |ψ = α |0 + β |1 then after measurement the state of the system collapses to |0 with probability |α|2 or to |1 with probability |β|2 . Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Hence, only one classical bit of information can be stored in one qubit. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 42 / 54
- 93. What is Superdense Coding? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Superdense Coding 5 is a protocol proposed by Charles H. Bennett and Stephen J. Wiesner. Introduction EPR Paradox Bell’s Theorem This is a simple protocol which enables the transportation of 2 cbits using only one ebit 6 . Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference 5 PRL 1992 Vol 69 Number 20 we shall be using ”cbit” for classical bit and ”ebit” for entangled bit 6 Henceforth, Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 43 / 54
- 94. Superdense Coding: Initial Setup Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha The initial step involves preparing an Entangled State. Alice and Bob prepare a Bell State, say Outline Introduction EPR Paradox |φ+ = 1 √ (|00 2 + |11 ) Bell’s Theorem Mathematical Notation Quantum Gates After preparation, Alice keeps one of the two entangled qubits with herself and sends the other one to Bob. Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 44 / 54
- 95. How to prepare the Bell State Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Before proceeding to the main protocol, the question that needs to be asked is - Mathematical Notation Quantum Gates Super Dense Coding How to prepare the Bell State? Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 45 / 54
- 96. How to prepare the Bell State Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Before proceeding to the main protocol, the question that needs to be asked is - Mathematical Notation Quantum Gates Super Dense Coding How to prepare the Bell State? Next Presentation Reference Consider Alice and Bob both start with one qubit each, both in state |0 . Alice operates her qubit only with a Hadamard Gate. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 45 / 54
- 97. How to prepare the Bell State Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Before proceeding to the main protocol, the question that needs to be asked is - Mathematical Notation Quantum Gates Super Dense Coding How to prepare the Bell State? Next Presentation Reference Consider Alice and Bob both start with one qubit each, both in state |0 . Alice operates her qubit only with a Hadamard Gate. This operation is followed by a CNOT Gate on the two qubits. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 45 / 54
- 98. How to prepare the Bell State Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline What happens when Alice operates a Hadamard Gate on her qubit? Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 46 / 54
- 99. How to prepare the Bell State Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline What happens when Alice operates a Hadamard Gate on her qubit? Introduction EPR Paradox Bell’s Theorem Mathematical Notation H |0 = Quantum Gates Super Dense Coding 1 √ 2 1 √ 2 Ritajit Majumdar, Arunabha Saha (CU) 1 √ 2 1 − √2 1 = 0 1 √ 2 1 √ 2 Introduction to Quantum Computation Next Presentation Reference September 25, 2013 46 / 54
- 100. How to prepare the Bell State Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline What happens when Alice operates a Hadamard Gate on her qubit? Introduction EPR Paradox Bell’s Theorem Mathematical Notation H |0 = Quantum Gates Super Dense Coding 1 √ 2 1 √ 2 = 1 √ 2 Ritajit Majumdar, Arunabha Saha (CU) 1 √ 2 1 − √2 1 = 0 1 √ 2 1 √ 2 Next Presentation Reference 1 1 Introduction to Quantum Computation September 25, 2013 46 / 54
- 101. Introduction to Quantum Computation How to prepare the Bell State Ritajit Majumdar, Arunabha Saha Outline What happens when Alice operates a Hadamard Gate on her qubit? Introduction EPR Paradox Bell’s Theorem Mathematical Notation H |0 = Quantum Gates Super Dense Coding 1 √ 2 1 √ 2 = 1 √ 2 Ritajit Majumdar, Arunabha Saha (CU) 1 √ 2 1 − √2 1 = 1 1 √ 2 1 = 0 1 + 0 1 √ 2 1 √ 2 1 √ 2 Next Presentation Reference 0 1 Introduction to Quantum Computation September 25, 2013 46 / 54
- 102. Introduction to Quantum Computation How to prepare the Bell State Ritajit Majumdar, Arunabha Saha Outline What happens when Alice operates a Hadamard Gate on her qubit? Introduction EPR Paradox Bell’s Theorem Mathematical Notation H |0 = Quantum Gates Super Dense Coding 1 √ 2 1 √ 2 = 1 √ 2 1 √ 2 1 − √2 1 = 1 = Ritajit Majumdar, Arunabha Saha (CU) 1 √ 2 1 = 0 1 √ 2 |0 + 1 + 0 1 √ 2 1 √ 2 1 √ 2 1 √ 2 Next Presentation Reference 0 1 |1 Introduction to Quantum Computation September 25, 2013 46 / 54
- 103. How to prepare the Bell State? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Now we have Alice’s qubit as 1 √ 2 |0 + EPR Paradox 1 √ 2 |1 while Bob’s qubit is |0 as before. Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 47 / 54
- 104. How to prepare the Bell State? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Now we have Alice’s qubit as 1 √ 2 |0 + EPR Paradox 1 √ 2 |1 Bell’s Theorem Mathematical Notation Quantum Gates while Bob’s qubit is |0 as before. Super Dense Coding So the 2 qubit system takes the form - Next Presentation Reference 1 √ 2 Ritajit Majumdar, Arunabha Saha (CU) |00 + 1 √ 2 |10 Introduction to Quantum Computation September 25, 2013 47 / 54
- 105. How to prepare the Bell State? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Now we have Alice’s qubit as 1 √ 2 EPR Paradox |0 + 1 √ 2 |1 Bell’s Theorem Mathematical Notation Quantum Gates while Bob’s qubit is |0 as before. Super Dense Coding So the 2 qubit system takes the form - Next Presentation Reference 1 √ 2 |00 + 1 √ 2 |10 Now applying CNOT Gate, the resultant state is 1 √ (|00 2 + |11 ) which is the required Bell State. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 47 / 54
- 106. Super Dense Conding: The Protocol Alice wants to send 2 cbits of information to Bob using her qubit. Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 48 / 54
- 107. Super Dense Conding: The Protocol Alice wants to send 2 cbits of information to Bob using her qubit. Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction There are 4 possible combinations of the 2 classical bits that Alice wants to send - EPR Paradox Bell’s Theorem Mathematical Notation 00, 01, 10, 11 Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 48 / 54
- 108. Super Dense Conding: The Protocol Alice wants to send 2 cbits of information to Bob using her qubit. Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction There are 4 possible combinations of the 2 classical bits that Alice wants to send - EPR Paradox Bell’s Theorem Mathematical Notation 00, 01, 10, 11 Quantum Gates Super Dense Coding Depending on which combination Alice wants to send, she performs the following operations on her qubit - Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Next Presentation Reference September 25, 2013 48 / 54
- 109. Super Dense Conding: The Protocol Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem After performing the required operation, Alice sends her qubit to Bob. Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 49 / 54
- 110. Super Dense Conding: The Protocol Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem After performing the required operation, Alice sends her qubit to Bob. Mathematical Notation Quantum Gates Super Dense Coding Bob now has both the qubits with him. Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 49 / 54
- 111. Super Dense Conding: The Protocol Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem After performing the required operation, Alice sends her qubit to Bob. Mathematical Notation Quantum Gates Super Dense Coding Bob now has both the qubits with him. Next Presentation Reference Bob performs the initial operations in reverse order, i.e. ﬁrst CNOT gate and then Hadamard Gate on the 1st qubit only. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 49 / 54
- 112. Super Dense Conding: The Protocol Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem After performing the required operation, Alice sends her qubit to Bob. Mathematical Notation Quantum Gates Super Dense Coding Bob now has both the qubits with him. Next Presentation Reference Bob performs the initial operations in reverse order, i.e. ﬁrst CNOT gate and then Hadamard Gate on the 1st qubit only. Bob now has the two classical bits that Alice sent her. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 49 / 54
- 113. Super Dense Conding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Consider Alice wants to send cbit 01 to Bob. Then she operates her qubit with the Pauli matrix σx or the bit ﬂip gate. Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference 7 NOTE: Alice can perform operation on her qubit only. Hence, the operation aﬀects only the 1st qubit. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 50 / 54
- 114. Super Dense Conding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Consider Alice wants to send cbit 01 to Bob. Then she operates her qubit with the Pauli matrix σx or the bit ﬂip gate. Bell’s Theorem So the state of the system after operation is: Super Dense Coding Mathematical Notation Quantum Gates Next Presentation σx |φ+ = 1 √ (|10 2 + |01 7 ) Reference 7 NOTE: Alice can perform operation on her qubit only. Hence, the operation aﬀects only the 1st qubit. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 50 / 54
- 115. Super Dense Conding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Consider Alice wants to send cbit 01 to Bob. Then she operates her qubit with the Pauli matrix σx or the bit ﬂip gate. Bell’s Theorem So the state of the system after operation is: Super Dense Coding Mathematical Notation Quantum Gates Next Presentation σx |φ+ = 1 √ (|10 2 + |01 7 ) Reference Alice now sends her qubit to Bob who has the other half of the entangled qubit. 7 NOTE: Alice can perform operation on her qubit only. Hence, the operation aﬀects only the 1st qubit. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 50 / 54
- 116. Super Dense Coding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Bob now performs CNOT operation on the system 1 √ (|10 + |01 ). The resultant state will be 2 Outline Introduction EPR Paradox 1 √ (|11 2 + |01 ) Bell’s Theorem Mathematical Notation Quantum Gates Super Dense Coding Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 51 / 54
- 117. Super Dense Coding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Bob now performs CNOT operation on the system 1 √ (|10 + |01 ). The resultant state will be 2 Outline Introduction EPR Paradox 1 √ (|11 2 + |01 ) Bell’s Theorem Mathematical Notation Now, the Hadamard Gate is applied on the 1st qubit only. We know - Quantum Gates Super Dense Coding Next Presentation 1 H |0 = √2 (|0 + |1 ) 1 and H |1 = √2 (|0 − |1 ) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 25, 2013 51 / 54
- 118. Super Dense Coding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Bob now performs CNOT operation on the system 1 √ (|10 + |01 ). The resultant state will be 2 Outline Introduction EPR Paradox 1 √ (|11 2 + |01 ) Bell’s Theorem Mathematical Notation Now, the Hadamard Gate is applied on the 1st qubit only. We know - Quantum Gates Super Dense Coding Next Presentation 1 H |0 = √2 (|0 + |1 ) 1 and H |1 = √2 (|0 − |1 ) Reference So the ﬁnal state will be - Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 51 / 54
- 119. Super Dense Coding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Bob now performs CNOT operation on the system 1 √ (|10 + |01 ). The resultant state will be 2 Outline Introduction EPR Paradox 1 √ (|11 2 + |01 ) Bell’s Theorem Mathematical Notation Now, the Hadamard Gate is applied on the 1st qubit only. We know - Quantum Gates Super Dense Coding Next Presentation 1 H |0 = √2 (|0 + |1 ) 1 and H |1 = √2 (|0 − |1 ) Reference So the ﬁnal state will be 1 1 √ ( √ (|0 2 2 − |1 ) |1 + Ritajit Majumdar, Arunabha Saha (CU) 1 √ (|0 2 + |1 ) |1 ) Introduction to Quantum Computation September 25, 2013 51 / 54
- 120. Super Dense Coding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Bob now performs CNOT operation on the system 1 √ (|10 + |01 ). The resultant state will be 2 Outline Introduction EPR Paradox 1 √ (|11 2 + |01 ) Bell’s Theorem Mathematical Notation Now, the Hadamard Gate is applied on the 1st qubit only. We know - Quantum Gates Super Dense Coding Next Presentation 1 H |0 = √2 (|0 + |1 ) 1 and H |1 = √2 (|0 − |1 ) Reference So the ﬁnal state will be 1 1 √ ( √ (|0 2 2 − |1 ) |1 + 1 √ (|0 2 + |1 ) |1 ) = 1 (|01 − |11 + |01 + |11 ) 2 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 51 / 54
- 121. Super Dense Coding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Bob now performs CNOT operation on the system 1 √ (|10 + |01 ). The resultant state will be 2 Outline Introduction EPR Paradox 1 √ (|11 2 + |01 ) Bell’s Theorem Mathematical Notation Now, the Hadamard Gate is applied on the 1st qubit only. We know - Quantum Gates Super Dense Coding Next Presentation 1 H |0 = √2 (|0 + |1 ) 1 and H |1 = √2 (|0 − |1 ) Reference So the ﬁnal state will be 1 1 √ ( √ (|0 2 2 − |1 ) |1 + 1 √ (|0 2 + |1 ) |1 ) = 1 (|01 − |11 + |01 + |11 ) 2 = 1 (2 |01 ) 2 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 51 / 54
- 122. Super Dense Coding: Case Study Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Bob now performs CNOT operation on the system 1 √ (|10 + |01 ). The resultant state will be 2 Outline Introduction EPR Paradox 1 √ (|11 2 + |01 ) Bell’s Theorem Mathematical Notation Now, the Hadamard Gate is applied on the 1st qubit only. We know - Quantum Gates Super Dense Coding Next Presentation 1 H |0 = √2 (|0 + |1 ) 1 and H |1 = √2 (|0 − |1 ) Reference So the ﬁnal state will be 1 1 √ ( √ (|0 2 2 − |1 ) |1 + 1 √ (|0 2 + |1 ) |1 ) = 1 (|01 − |11 + |01 + |11 ) 2 = 1 (2 |01 ) 2 = |01 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 51 / 54
- 123. Coming up in next talk... Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox Bell’s Theorem Mathematical Notation No Cloning Theorem Quantum Gates Quantum Teleportation Super Dense Coding Next Presentation Conclusive Quantum Teleportation Reference Quantum Algorithms and many more... Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 52 / 54
- 124. Introduction to Quantum Computation Reference Ritajit Majumdar, Arunabha Saha Michael A. Nielsen, Isaac Chuang Quantum Computation and Quantum Information Cambridge University Press Outline David J. Griﬃths Introduction to Quantum Mechanics Prentice Hall, 2nd Edition Mathematical Notation Umesh Vazirani, University of California Berkeley Quantum Mechanics and Quantum Computation https://class.coursera.org/qcomp-2012-001/ Reference Introduction EPR Paradox Bell’s Theorem Quantum Gates Super Dense Coding Next Presentation Michael A. Nielsen, University of Queensland Quantum Computing for the determined http://michaelnielsen.org/blog/ quantum-computing-for-the-determined/ www.springer.com Quantum Gates Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 53 / 54
- 125. Introduction to Quantum Computation Reference Ritajit Majumdar, Arunabha Saha Outline Introduction EPR Paradox A. Einstein, B. Podolsky, N. Rosen, Physical Review 47, 777 (1935) Bell’s Theorem Mathematical Notation Quantum Gates J.S. Bell Physics 1, 195 (1964) Super Dense Coding Next Presentation Reference D. Bohm Physical Review 85, 166, 180 (1952) Charles H. Bennett, Stephen J. Wiesner Physical Review Letters, Nov 16, 1992 Volume 69, Number 20 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 54 / 54

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